
\noindent\underline{\em Proof of Lemma~\ref{lem:br}.}
The feasible power set of the $i$-th femto AP is convex and
$U_{i}(p_{i},p_{-i})$ is a strict concave function of $p_{i}$ , since,
for proportional fair $(\alpha = 1)$
\begin{eqnarray} \frac{\partial^{2}U_{i}({\bf p})}{\partial
p_{i}^{2}}&=&-\frac{N_{i}h_{ii}^{2}(1+\log(1+\frac{h_{ii}p_i}{I_{t}}))}{D^{2}}<0,
\end{eqnarray}
and for all $\alpha \neq 1,$
%=======================================
\begin{equation}
\frac{\partial^{2}U_{i}({\bf p})}{\partial
p_{i}^{2}}=-\frac{(N_{i})^{\alpha}h_{ii}^{2}
(\log(1+\frac{h_{ii}p_i}{I_{t}}) + \alpha)}{(I_{t}
+h_{ii}p_{i})^{2}(\log(1+\frac{h_{ii}p_i}{I_{t}}))^{\alpha+1}} <0.
\end{equation}
%=======================================
Thus, the optimal point is unique.
\QED

\vspace{1cm}
\smallskip
\noindent\underline{\em Proof of Lemma~\ref{lem:scal}.}
Let ${\bf p}_{max}$ be the power vector where all elements are $p_{max}.$
\begin{compactenum}[\em i)]
\item $\beta {\bf B}({\bf p}) > {\bf p}_{max}$

In this case, trivially, $\beta {\bf B}({\bf p}) > {\bf B}(\beta
{\bf p})$, because $\beta {\bf B}({\bf p}) > P_{max} \ge {\bf B}(\beta
{\bf p})$

\item $\beta {\bf B}({\bf p}) \le P_{max}$


As shown in {\em Proof of Lemma~\ref{lem:br},} the utility function
$U_{i}({\bf p})$ is a strict concave function for
$p_i$. Moreover, the function of $p_i$, $U_{i}(p_{i},\beta p_{-i})$, is decreasing at
the point $p_{i}=\beta B_{i}(p_{-i})$ , because
\begin{eqnarray*}
\frac{\partial U_{i}(p_{i},\beta p_{-i})}{\partial p_{i}}|_{p_{i}=\beta B_{i}(p_{-i})}
&=&\frac{N_{i}h_{ii}}{(I+\beta\sum_{j\neq i}h_{ji}p_{j}+\beta h_{ii}B_{i}(p_{-i}))(\log(1+\frac{\beta h_{ii}B_{i}(p_{-i})}{I+\beta\sum_{j\neq i}h_{ji}p_{j}}))^{\alpha}}-\beta\\
&<&\frac{N_{i}h_{ii}}{(I+\sum_{j\neq i}h_{ji}p_{j}+h_{ii}B_{i}(p_{-i}))(\log(1+\frac{h_{ii}B_{i}(p_{-i})}{I+\sum_{j\neq i}h_{ji}p_{j}}))^{\alpha}}-\beta=0.\end{eqnarray*}
Therefore, $B_{i}(\beta p_{-i})<\beta B_{i}(p_{-i})$.
\end{compactenum}

Thus, from (i) and (ii), $\beta {\bf B}({\bf p}) > {\bf B}(\beta {\bf p})$ for all $\beta > 1$.
\QED
\vspace{1cm}

\smallskip
\noindent\underline{\em Proof of Theorem~\ref{thm:uni}.}
NEs can be defined as power vectors that satisfy ${\bf p}={\bf
B}({\bf p}).$ Since the game is supermodular, at least
one NE exists, and, due to the {\em Scalability}, there is a
unique fixed point (NE) from Theorem 1 in \cite{Y95FUPC}.
Therefore, the
unique NE is globally stable.
\QED
\vspace{1cm}
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